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5.3 The Central Limit Theorem •Roll a die 5 times and record the value of each roll. •Find the mean of the values of the 5 rolls. •Repeat this 250 times. x=3.504 s=.7826 n=5 •Roll a die 10 times and record the value of each roll. •Find the mean of the values of the 10 rolls •Repeat this 250 times. Poll: Toss a die 10 times and record your resu... x=3.48 s=.5321 n=10 •Roll a die 20 times. •Find the mean of the values of the 20 rolls. •Repeat this 250 times. x=3.487 s=.4155 n=20 What do you notice about the shape of the distribution of sample means? Central Limit Theorem • Suppose we take many random samples of size n for a variable with any distribution--For large sample sizes: 1. The distribution of means will be approximately a normal distribution. 1, 2, 3, 4, 5, 6 • Mean: =3.5 • Standard Deviation: =1.7078 • How does the mean of the sample means compare to the mean of the population? • Remember for 250 trials: • When n=5, x=3.504 • When n=10, x=3.48 • When n=20, x=3.487 • How does the mean of the sample means compare to the mean of the population? Central Limit Theorem • Suppose we take many random samples of size n for a variable with any distribution--For large sample sizes: 1. The distribution of means will be approximately a normal distribution. 2. The mean of the distribution of means approaches the population mean, . 1, 2, 3, 4, 5, 6 • Mean: =3.5 • Standard Deviation: =1.7078 • How does the standard deviation of the sample means compare to the standard deviation of the population? • Remember for 250 trials: • When n=5, s=.7826 • When n=10, s=.5321 • When n=20, s=.4155 • How does the standard deviation of the sample means compare to the standard deviation of the population? Central Limit Theorem • Suppose we take many random samples of size n for a variable with any distribution--For large sample sizes: 1. The distribution of means will be approximately a normal distribution. 2. The mean of the distribution of means approaches the population mean, . 3. The standard deviation of the distribution of means approaches . n Cost of owning a dog • Suppose that the average yearly cost per household of owning dog is $186.80 with a standard deviation of $32. Assume many samples of size n are taken from a large population of dog owners and the mean cost is computed for each sample. • If the sample size is n=25, find the mean and standard deviation of the sample means. • If the sample size is n=100, find the mean and standard deviation of the sample means. Teacher’s salary • The average teacher’s salary in New Jersey (ranked first among states) is $52,174. Suppose the distribution is normal with standard deviation equal to $7500. • What percentage of individual teachers make less than $45,000? • Assume a random sample of 64 teachers is selected, what percentage of the sample means is a salary less than $45,000? Height of basketball players • Assume the heights of men are normally distributed with a mean of 70.0 inches and a standard deviation of 2.8 inches. • What percentage of individual men have a height greater than 72 inches? • The mean height of a 16 man roster on a high school team is at least 72 inches. What percentage of sample means from a sample of size 16 are greater than 72 inches? • Is this basketball team unusually tall?